In this work, we apply ASP to HDG schemes for three different elliptic
problems with a parameter dependent low order term.
In particular, we consider a symmetric interior penalty HDG scheme for the
scalar reaction-diffusion equation, a divergence-conforming HDG scheme
for a vectorial reaction-diffusion equation, and a \(C^0\) -continuous interior
penalty HDG scheme for the generalized biharmonic equation with a low order term.
Uniform preconditioners are obtained for each case and the general ASP theory
[1]} is used to prove the optimality with respect to the mesh size and
uniformity with respect to the low order parameter. We do not theoretically
address the issue of robustness of the preconditioner with respect to the polynomial
degree, but our numerical results indicate the growth of the iteration counts of
the preconditioned conjugate gradient (PCG) algorithm on the polynomial degree is quite mild.
We point out that while our analysis is focused on interior penalty type HDG schemes
for each problem, our results are easily applicable to other HDG formulations and other
hybrid finite element methods.
